Definition:One-Sided Derivative
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Definition
A one-sided derivative is a right-hand derivative or a left-hand derivative.
Right-Hand Derivative
Let $f: \R \to \R$ be a real function.
The right-hand derivative of $f$ is defined as the right-hand limit:
- $\ds \map {f'_+} x = \lim_{h \mathop \to 0^+} \frac {\map f {x + h} - \map f x} h$
If the right-hand derivative exists, then $f$ is said to be right-hand differentiable at $x$.
Left-Hand Derivative
Let $f: \R \to \R$ be a real function.
The left-hand derivative of $f$ is defined as the left-hand limit:
- $\ds \map {f'_-} x = \lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h$
If the left-hand derivative exists, then $f$ is said to be left-hand differentiable at $x$.
Examples
Absolute Value Function at $x = 0$
Let $f$ be the real function defined as:
- $\map f x = \size x$
where $\size x$ denotes the absolute value function.
Then:
\(\ds \map {f'_+} 0\) | \(=\) | \(\ds 1\) | where $\map {f'_+} 0$ denotes the right-hand derivative of $f$ at $x = 0$ | |||||||||||
\(\ds \map {f'_-} 0\) | \(=\) | \(\ds -1\) | where $\map {f'_-} 0$ denotes the left-hand derivative of $f$ at $x = 0$ |
while the derivative of $f$ at $x = 0$ does not exist.